We study the problem of detecting the presence of an underlying
high-dimensional geometric structure in a random graph. Under the null
hypothesis, the observed graph is a realization of an Erd\H{o}s-R\'enyi random
graph G(n,p). Under the alternative, the graph is generated from the
G(n,p,d) model, where each vertex corresponds to a latent independent random
vector uniformly distributed on the sphere Sd−1, and two vertices
are connected if the corresponding latent vectors are close enough. In the
dense regime (i.e., p is a constant), we propose a near-optimal and
computationally efficient testing procedure based on a new quantity which we
call signed triangles. The proof of the detection lower bound is based on a new
bound on the total variation distance between a Wishart matrix and an
appropriately normalized GOE matrix. In the sparse regime, we make a conjecture
for the optimal detection boundary. We conclude the paper with some preliminary
steps on the problem of estimating the dimension in G(n,p,d).Comment: 28 pages; v2 contains minor change